Optimal geodesics for boundary points of the Gardiner-Masur compactification

The Gardiner-Masur compactification of Teichmüller space is homeomorphic to the horofunction compactification of the Teichmüller metric. Let $\xi$ and $\eta$ be a pair of boundary points in the Gardiner-Masur compactification that fill up the surface. We show that there is a unique Teichmüller geodesic which is optimal for the horofunctions corresponding to $\xi$ and $\eta$. In particular, when $\xi$ and $\eta$ are Busemann points that fill up the surface, the geodesic converges to $\xi$ in forward direction and to $\eta$ in backward direction. As an application, we show that if ${\mathbf{G}}_n$ is a sequence of Teichmüller geodesics passing through $X_n$ and $Y_n$ such that $X_n\to \xi$ and $Y_n\to \eta$, then ${\mathbf{G}}_n$ converges to a unique Teichmüller geodesic.